Weighted norm inequalities for maximal functions and singular integrals

Coifman, Ronald R.; Fefferman, Charles

doi:10.4064/sm-51-3-241-250

Cited by 888 publications

(638 citation statements)

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“…123-124], where we can find the case when m is homogeneous of degree 0. Also, for a positive integer k, the derivatives of m k j satisfy [4,14]). Thus, by the evaluation of C j,k,M we have…”

Section: Invertibility Of Homogeneous Fourier Multipliers and Littlewmentioning

confidence: 99%

Littlewood–Paley Equivalence and Homogeneous Fourier Multipliers

Sato

2016

Integr. Equ. Oper. Theory

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Abstract. We consider certain Littlewood-Paley operators and prove characterization of some function spaces in terms of those operators. When treating weighted Lebesgue spaces, a generalization to weighted spaces will be made for Hörmander's theorem on the invertibility of homogeneous Fourier multipliers. Also, applications to the theory of Sobolev spaces will be given. IntroductionWe consider the Littlewood-Paley function on R n defined by, where ψ t (x) = t −n ψ(t −1 x). The following result of Benedek, Calderón and Panzone [2] on the L p boundedness, 1 < p < ∞, of g ψ is well-known.Theorem A. We assume (1.1) for ψ andwhereBy the Plancherel theorem, it follows that g ψ is bounded on L 2 (R n ) if and only if m ∈ L ∞ (R n ), where m(ξ) = ∞ 0 |ψ(tξ)| 2 dt/t, which is a homogeneous function of degree 0. Here the Fourier transform is defined aŝ ψ(ξ) = R n ψ(x)e −2πi x,ξ dx, x, ξ = n k=1x k ξ k .

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Section: Invertibility Of Homogeneous Fourier Multipliers and Littlewmentioning

confidence: 99%

Littlewood–Paley Equivalence and Homogeneous Fourier Multipliers

Sato

2016

Integr. Equ. Oper. Theory

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“…We shall need the following estimate due to R. Coifman [2,3]: for any 0 < p < ∞ and for w ∈ A ∞ , (2.8)…”

Section: 2mentioning

confidence: 99%

Weak Type Estimates for Singular Integrals Related to a Dual Problem of Muckenhoupt–Wheeden

Lerner

Ombrosi

Pérez

2008

J Fourier Anal Appl

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Abstract.A well known open problem of Muckenhoupt-Wheeden says that any Calderón-Zygmund singular integral operator T is of weak type (1, 1) with respect to a couple of weights (w, M w). In this paper we consider a somewhat "dual" problem:We prove a weaker version of this inequality with M 3 w instead of M w. Also we study a related question about the behavior of the constant in terms of the A 1 characteristic of w.

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“…Note that if F is a constant function, then our weights satisfy a reverse Hölder inequality and therefore they belong to the usual Muckenhaupt class A ∞ (see [3] and [2]). …”

Section: Then There Exists a Disjoint (At Most Countable) Subfamily {B(xmentioning

confidence: 99%